Exact Diagonalization of Few-electron Quantum Dots

Exact Diagonalization of Few- electron Quantum Dots

Shirin Hakimi

Degree project work in physics Level: C

No: 2009:FY2

University of Kalmar

Degree project works made at the University of Kalmar, School of Pure and Applied Natural Sciences,

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Exact Diagonalization of Few-electron Quantum Dots

Shirin Hakimi

Examination Project Work in Physics, 15 ECTS credits

Supervisors: Professor Carlo Maria Canali Kalmar University Examiner: Associate Professor Magnus Paulsson Kalmar University

Abstract

We consider a system of few electrons trapped in a two-dimensional circular

quantum dot with harmonic confinement and in the presence of a

homogeneous magnetic field, with focus on the role of e-e interaction. By

performing the exact diagonalization of the Hamiltonian in second quantization,

the low-lying energy levels for spin polarized system are obtained. The singlet-

triplet oscillation in the ground state of the two-electron system showing up in

the result is explained due to the role of Coulomb interaction. The splitting of

the lowest Landau level is another effect of the e-e interaction, which is also

observed in the results.

Acknowledgments

I would like to thank my supervisor Professor Carlo Maria Canali for suggesting me such an interesting subject, for his help in scientific as well as practical matters, and for the interesting discussions that we had. I really enjoyed doing this project. I also want to thank Professor Stephanie Reimann for sharing her software, Associate Professor Magnus Paulsson for his help with computer related issues, and Mr. Javier Francisco Nossa for helping me with Latex.

Contents

1 Introduction 5

2 Background 6

2.1 Single particle in a harmonic confinement and a homogeneous

magnetic field . . . 6

2.2 Interacting many-particle system . . . 8

2.2.1 Second quantization . . . 9

2.2.2 Exact diagonalization . . . 12

2.2.3 Hartree-Fock Approximation . . . 14

2.2.4 Analytical solution . . . 15

2.3 On the importance of e-e Coulomb interaction . . . 16

2.3.1 Wigner Crystal . . . 16

2.3.2 Fractional quantum Hall effect . . . 17

3 Model and Implementation 18 4 Numerical results and discussion 19 4.1 Two-electron system . . . 19

4.2 Few-electron system . . . 20

5 Conclusion 21

1 Introduction

A quantum dot (QD) is a nanoscaled confinement of carriers, which shows very strong quantum mechanical properties due to its nanoscaled size. One of these properties is the optical property of QD, which has been used by stained glass makers a few centuries ago by confining different size of gold quantum dots in glass to provide different colors. More consciously, in 1980, L. Brus fabricated semiconductor quantum dot at Bell Lab using semiconductors as confinement.

The name, quantum dot, is used for the first time by M. Reed in 1985, who defined it as “a completely spatially quantized system where the carriers have zero degree of freedom” [12]. The nanosize of quantum dots leads to the discreteness of energy levels as atoms. The energy levels can be designed by tuning the QD parameters. Due to this fact, quantum dot got its second name as the artificial atom by P.A. Maksym and T. Chakraborty [8].

The theory behind QD is almost as old as the history of quantum mechanics.

In 1928, V.A. Fock presents the energy spectrum of one electron trapped in a two-dimensional (2-d) circular QD with harmonic confinement and an external homogeneous magnetic field [5]. Independently, in 1930, C.G. Darwin presents same solution [3].

To take the next step and add more electrons to the system gives rise to the practical difficulties in solving the system. The traditional way of solving the Schr¨odinger equation is impossible for more than two electrons, due to the mix- ing term coming from the e-e Coulomb interaction. The approximation methods like perturbation theory and the Hartree-Fock method have been used widely, but it turns out that the energy of e-e interaction is considerable large, and the approximation methods cannot reveal the effect of e-e Coulomb interaction in the energy spectrum [10].

One way to obtain the exact solution is to solve the Schr¨odinger equation analytically, which can be possible just for two electrons. This has been done for a 2-d circular quantum dot confining two electrons in the center of mass coordinate system [14]. Another approach for the exact solution of few-electron systems is exact diagonalization of the Hamiltonian matrix. In this method, the wavefunction of interacting system is written as the linear combination of the Slater determinants constructed by single-particle wavefunctions. The diagonalization of Hamiltonian in this basis results in the energy spectrum and the eigenfunctions of the system.

Exact diagonalization calculation for a 2-electron quantum dot with infinite well potential was performed by G.W. Bryant in 1987 [1]. The spin polarized quantum dot containing up to 10 electrons with harmonic confinement in the presence of a homogeneous magnetic field has been studied [8]. In 1993, D.

Pfannkuche, et al. reported their studies on the same system with two electrons, which they called it quantum helium [10].

In continuation, two strategies have been followed. One is making the model more and more accurate by taking into account the ignored factors like spin- orbit interaction [11] or anharmonic confinement. The other strategy is studying the response of the system to the additional external factors like inhomogeneous

magnetic field, external electrical field, rotating the system [9] or different shape of quantum dot, elliptical and stadium shape.

A few-electron QD with harmonic confinement in a homogeneous magnetic field can reveal interesting effects, due to the e-e interaction. Among these effects, Wigner crystallization [16] at low electronic density and the fractional quantum Hall effect(FQHE) at high magnetic field can be mentioned. Addition spectrum (the energy to add one electron to a QD) for few-electron quantum dot in the fractional quantum Hall effect regime has been studied by S.-R.E Yang et al. by using the exact diagonalization method [17].

In this thesis after a review of the related background, we report our numer- ical results and make a qualitative discussion about the results.

2 Background

2.1 Single particle in a harmonic confinement and a ho- mogeneous magnetic field

A simple model for a 2-d circular QD containing one electron in the presence of a homogeneous magnetic field can be introduced as follows,

H = (P +^e_cA)² 2m^∗ +1

2m^∗ω₀²r², (1)

where the QD confinement is modeled as harmonic potential.

By using the symmetric gauge A = ¹₂B×r, the Hamiltonian can be rewritten as,

H = P²

2m^∗ − m^L.B + e²

8m^∗c²(B × r)²+1

2m^∗ω²₀r², (2) where mL = −_2m^e∗cL. It is also useful to write the Hamiltonian in terms of cyclotron frequency ωc= eB/m^∗c,

H = P² 2m^∗−1

2ωcL+1 2m^∗(ω²_c

4 + ω₀²)r². (3) To obtain the energy spectrum and the wave functions of this system, one can solve the Schr¨odinger equation in the polar coordinates,

− ¯h² 2m^∗

 ∂²ψ

∂r² +1 r

∂ψ

∂r + 1 r²

∂²ψ

∂θ²



−i¯hωc

2

∂ψ

∂θ +m^∗ 2 (ω²_c

4 + ω₀²)r²ψ = Eψ . (4) Since the variables are separable and the system is symmetric around z axis, one can consider a solution like ψ = f (r)e^−ilθ.

Using the notation of magnetic length

lB=

 ¯h eB

¹₂

, (5)

and the dimensionless variable b, defined as below,

b =



1 + 4ω₀² ω_c²

¹₂

, (6)

the eigenfunctions can be written as,

ψnl(r, θ) = exp(−ilθ)√ 2π

 n!

2^|l|(n + |l|)!

¹₂ (

√b

lB) exp(−br² 4l²_B)(

√br lB

)^|l|L^|l|_n  br² 2l²_B

 , (7) where L^|l|n is the Laguerre polynomial, which comes from the harmonic con- finement and modifies the tail of the wave function compared with the wave function in an infinite well confinement.

The eigenvalues are,

E = (2n + |l| + 1)¯h(ω0²+ω²c

4 )¹² −1

2¯hωcl (8)

= (2n + |l| + 1)¯hΩ − 1

2¯hωcl n = 0, 1, 2, . . . l = 0, ±1, ±2, . . . ,

which is known as Fock-Darwin spectrum, Fig. 1. It is useful to study two limit cases:

1. When the magnetic field is zero (B=0). In this case the energy spectrum is reduced to,

E = (2n + |l| + 1)¯hω0 (9)

, which leads to the shell structure of QD.

Subshell (n, l) Energy Degeneracy

1 (0, 0) ¯hω0 2

2 (0, 1)(0, −1) 2¯hω0 4

3 (1, 0)(0, 2)(0, −2) 3¯hω0 6

4 (1, 1)(1, −1)(0, 3)(0, −3) 4¯hω0 8 5 (2, 0)(1, 2)(1, −2)(0, 4)(0, −4) 5¯hω0 10

In analogy to atom, the number of degeneracy for the subshells (2,4,6,8,10,...) is called magic numbers.

2. When the magnetic field is very strong B ≫ 1.

In this case ωc≫ ω⁰, so we can ignore the confinement potential. The energy spectrum will be,

E =¯h

2(2n + |l| − l + 1)ω^c. (10)

The energy levels which are known as Landau levels will be,

Landau level (n, l) Energy 0 (0, l ≥ 0) ¹2¯hωc

1 (1, l ≥ 0) ³2¯hωc

2 (2, l ≥ 0) ⁵2¯hωc

3 (3, l ≥ 0) ⁷2¯hωc

As we see, the quantum number n can label the Landau levels. The degen- eracy for each Landau level can be calculated as,

D = 2Φ Φ0

, (11)

where Φ0= hc/e is the flux quantum, and Φ = BA is the magnetic flux through the QD. By taking into account the Zeeman effect, we lose the spin degeneracy.

Degeneracy or the total states of the lowest Landau level will be reduced to,

D = Φ Φ0

, (12)

The expectation value of radius part of the wavefunction Eq. (7) of the Landau levels is,

R²_nl= 2λ²_B(2n + |l| + 1) , (13) where λ²_B =^l2B_b . This shows us the localization of the electrons in rings, namely Landau rings. By increasing the magnetic field, the Landau rings shrinks, and the effect of the e-e interaction plays a big role.

2.2 Interacting many-particle system

In general the Hamiltonian of an interacting many-particle system can be writ- ten as,

H = ˆˆ H⁽¹⁾+ ˆH⁽²⁾=

N

X

i=1

Tˆi+1 2

N

X

i6=j=1

Vˆij , (14)

where the first term is all single-particle operators containing the kinetic en- ergy and single-particle potential of the particles, and the second term is the interaction potential between particles.

Immediately, it can be seen that due to the interaction term in the Hamil- tonian, the variables are not separable, and we will face difficulties in the tra- ditional way of solving the Schr¨odinger differential equation. However, this is just possible for two-particle system by writing the Hamiltonian in the center of mass coordinate system. There are also approximation methods to treat the

many-particle Hamiltonian. One of these methods is the Hartree-Fock approx- imation which, is reviewed briefly in the next section. But the failure of the approximation methods requires the exact solution of many-particle systems.

One candidate for exact solution method is exact diagonalization of the Hamil- tonian in second quantization. This is the method that we used in this project and is explained in detail in this report.

Basically a many-particle problem in quantum mechanics is about occu- pying single-particle energy states by N identical particles. The states of an N -particle system are vector states of subspace of tensor product of N single- particle Hilbert space, which is called Fock space,

F_ν =

∞

M

N =0

SνHN

N . (15)

The function Sν symmetrizes the space at ν = 1 for bosons, and anti- symmetrizes the space at ν = −1 for fermions to satisfy the Pauli exclusion principle.

For fermions, one vector state of the Fock space can be written as the fol- lowing determinant,

|Ψi = 1

√N ! X

p

(−1)^δpP (|1q1i|2q2i . . . |NqNi) (16)

= 1

√N !         

|1q¹i |2q¹i . . . |Nq¹i

|1q²i |2q²i . . . |Nq²i ... ... . .. ...

|1qNi |2qNi . . . |NqNi          ,

which is called Slater determinant. P stands for permutation, the sign (−1)^δp= +1(−1) if the permutation is even (odd). By notation, for example |2q¹i, we mean that the particle number 2 is in the quantum state q1. A very transparent way of writing the Slater determinant is occupation number formalism,

|Ψi = |1, 1, 1, . . . , 1, 0, 0, 0, . . .i , (17) which presents a many-particle state, that the first N states are occupied by one fermion each, and the rest of states are empty.

All combinations of occupation of N states give a complete set of orthonormal basis for the Fock space. This consideration is useful, when we want to make Fock space basis for our system.

2.2.1 Second quantization

Where in first quantization, dynamical classical variables q and p in the Hamil- tonian are replaced by their operators, and satisfy the commutation realtion,

[ ˆqi, ˆpj] = i¯hδij [ ˆqi, ˆqj] = [ ˆpi, ˆpj] = 0 , (18) in second quantization, q and p are complex components. Instead, we deal with field operators, which are responsible for population and depopulation of states in the Fock space, defined as,

ψ(r) =ˆ X

i

φi(r)ci (19)

ψˆ^†(r) =X

i

φ^∗_i(r)c^†_i ,

where φ(r) is single-particle wave function, c and c^† are called creation and de- struction operator. Field operators satisfy the commutation relation for bosons and the anticommutation relation for fermions,

[ ˆψ(ri), ˆψ^†(rj)]∓= δij [ ˆψ(ri), ˆψ(rj)]∓ = [ ˆψ^†(ri), ˆψ^†(rj)]∓ = 0 . (20) Creation and destruction operators are originally borrowed from the quan- tum field theory, where they were responsible for creation and annihilation of a particle. In many-particle quantum mechanics, they are responsible for popula- tion and depopulation of a Fock state. However, the idea is that by using these operators (c and c^†), we use single-particle states to write the Hamiltonian in Fock basis.

Considering the single-particle operator of the many-particle Hamiltonian in first quantization, Eq. 14, we can write,

Tˆj = ˆIjTˆjIˆj=X

rt

|jrihjr| ˆTj|jtihjt| =X

rt

hjr| ˆTj|jti|jrihjt| =X

rt

Trt|jrihjt| , (21) where ˆIj is the unitary operator, and and |jri is the eigenstate |ri for the single- particle number j. Trt is the matrix elements of the single-particle operator, and it can be calculated by using the wavefunction in the configuration or the momentum space as,

Tst= hjs| ˆTj|jti = Z

d³rjψ^∗_s(rj) ˆT (rj)ψt(rj) , (22) where ψ(rj) is the single-particle wave function. The sum of single-particle operators can be written as,

H⁽¹⁾=

N

X

j=1

Tˆj=X

rt

Trt

X

j

|jrihjt| =X

rt

TrtErt. (23) To understand the role of operator Ert =P

j|jrihjt|, we study the effect of that on a many-particle state (Slater determinant).

Ert|Ψi = (X

j

|jrihjt|) 1

√N ! X

p

(−1)^δpP (|1q1i|2q2i . . . |αqα= αti . . . |NqNi|)

= 1

√N ! X

p

(−1)^δpP (|1q¹i|2q²i . . . |αri . . . |Nq^Ni), (24)

|Ψi = 1

√N ! X

p

(−1)^δpP (|1q1i|2q2i . . . |NqNi) (25)

= 1

√N !         

|1q¹i |2q¹i . . . |Nq¹i

|1q²i |2q²i . . . |Nq²i ... ... . .. ...

|1qNi |2qNi . . . |NqNi          .

To obtain this, we used the following relation,

|jrihjt|jsi = δts|jri . (26)

Example: Consider a 3-particle system, and choose r = q4 and t = q2.

Eq₄q₂|Ψi = Eq₄q₂( 1

√3!

|1q1i |2q1i |3q1i

|1q2i |2q2i |3q2i

|1q3i |2q3i |3q3i      

) = 1

√3!

|1q1i |2q1i |3q1i

|1q3i |2q3i |3q3i

|1q4i |2q4i |3q4i       .

In the occupation number formalism, it can be written as,

Eq₄q₂|1, 1, 1, 0, 0, 0, . . .i = |1, 0, 1, 1, 0, 0, . . .i , As we see, the operator Ert=P

j|jrihjt| removes t-state, and add r-state to the Slater determinant. In the occupation number formalism, we can claim that its effect is to depopulate the t-state, and populate the r-state. The behavior of Ert operator can be formulated by creation and annihilation (destruction) operators. As we know, the effect of the creation/destruction operator on a state is to increase/decrease a quanta. Since here we are dealing with the occupation of the states by fermions, the effect will be population/depopulation of the corresponding state. So, we can write,

Ert= c^†_rct. (27)

Then single-particle operator in the second quantization formalism will be,

Hˆ⁽¹⁾=

N

X

j=1

Tˆj=X

rt

Trtc^†_rct.(28)

With almost the same technique, we can find the two-particle operator in the second quantization as,

H⁽²⁾= 1 2

N

X

i6=j=1

Vˆij= 1 2

X

pqrs

Vpqrsc^†_pc^†_qcscr, (29) where Vpqrs is the matrix element of two-particle operator related to the inter- action term in the Hamiltonian and, it is known as interaction matrix elements.

It can be calculated as,

Vpqrs= hipjq|V^ij|irjsi = Z Z

d³rid³rjψ^∗_p(ri)ψ^∗_q(rj)V (ri, rj)ψr(ri)ψs(rj) , (30) where ψ(ri) and ψ(rj) are the single-particle wave functions.

Now, we can write the Hamiltonian in second quantization as, H = ˆˆ H⁽¹⁾+ ˆH⁽²⁾=X

rt

Trtc^†_rct+1 2

X

pqrs

Vpqrsc^†_pc^†_qcscr. (31) As we see, particle indices in the first quantization turn to indices of single- particle states in the second quantization. For constructing the Hamiltonian in second quantization, we need to calculate the matrix elements using Eq.(22) (30) and creation and destruction operators. Since creation/destruction operators act on many-particle wavefunction, and populate/depopulate a many-particle state, so they have many-particle basis. In addition, many-particle basis is the Fock space basis which can be constructed based on single-particle basis.

In this chapter, we followed the description in [13] and [4].

2.2.2 Exact diagonalization

One way of finding the eigenfunctions and the eigenvalues is diagonalization of the Hamiltonian in second quantization. To do this, we need to cut the single- particle Hilbert space into a finite space, which costs the reduction of precision of this method, particularly for big number of particles and high excited levels.

To approach the exact solution, in one hand, we can consider the states that con- tribute more in the energy values, i.e. the low lying single-particle eigenstates.

In the other hand, we can choose a large number of single-particle eigenstates, but we should be aware that by increasing the number of the single-particle eigenstates, the dimension of the Hamiltonian increases very rapidly. The num- ber of basis of the Hamiltonian or Slater determinant can be calculated as,

d = δ!

(δ − N)!N!, (32)

where δ is the number of single-particle eigenstates, and N is the number of electrons. For example, considering 36 single-particle states, the size of the

Hamiltonian for two, three and four-electron system are respectively 630, 14 280 and 353 430.

The diagonalization of the Hamiltonian in second quantization, Eq. (31), can be done by standard methods. One of the exact numerical diagonalization methods is Lanczos method. The idea behind this method is that by starting with a test vector and through a process of minimized iterations with least squares, we build a complete set of orthonormal vectors. The matrix in these basis will be tridiagonal, which is convenient for diagonalization. This method is particularly convenient for large spars matrices. Note that the smart choice of the test vector leads to the rapid convergence.

Here we follow the Lanczos method for a Hermitian matrix M [7]. First we choose a test vector |x0i and build up another vector,

|x1i = M|x0i − a0e^iα|x0i . (33) By minimizing the hx1|x1i, and using the hermitian property of Matrix M, we find the variational parameter a0 as,

a0=hx⁰|M|x⁰i

hx0|x0i . (34)

As we see, the new vector |x¹i is orthogonal to the original vector |x⁰i. We proceed by finding the third vector as,

|x²i = M|x¹i − a¹|x¹i − b¹|x⁰i . (35) By minimizing hx2|x2i, we obtain the parameters,

a1=hx¹|M|x¹i

hx1|x1i , (36)

b1= hx⁰|M|x¹i

hx0|x0i = hx¹|x¹i

hx0|x0i . (37)

It is easy to prove that this vector is orthogonal to the previous vectors. By repeating the same procedure, we will find a set of orthogonal vectors, defined as,

|xⁿ⁺¹i = M|xⁿi − aⁿ|xⁿi − bⁿ|xⁿ⁻¹i , (38) for n ≥ 0 and |x−1i = 0, and the coefficients are,

an= hxⁿ|M|xⁿi

hxn|xni n = 0, 1, 2, . . . , (39) bn =hxⁿ⁻¹|M|xⁿi

hxn−1|xn−1i = hxⁿ|xⁿi

hxn−1|xn−1i n = 1, 2, . . . . (40) Matrix M , in this basis is tridiagonal with diagonal elements an and off- diagonal elements bn. There are standard methods to diagonalize the tridiagonal matrix and find the eigenvectors and the eigenvalues.

2.2.3 Hartree-Fock Approximation

Hartree-Fock approximation is kind of mean field theory, where the interaction potential is reduced to a single-particle effective potential. In this way,a many- particle problem is reduced to a single-particle problem.

Consider the Hamiltonian of an interacting many-particle system in first quantization,

H =ˆ

N

X

i=1

Tˆi+1 2

N

X

i6=j=1

Vˆij , (41)

where the first term is the sum om all single-particle operators, and the second term is the sum of all interaction potential.

We can define the Hartree-Fock Hamiltonian by writing the interacting po- tential as effective single-particle potential,

HˆHF =

N

X

i=1

ˆti+ ˆv_HF⁽ⁱ⁾ 

, (42)

where ˆvHF is called Hartree-Fock potential. The aim is to solve the eigenvalue problem for this Hamiltonian,

HˆHFΦ = EΦ , (43)

where Φ is the Slater determinant. To solve this, we need to find the Hartree- Fock potential. We let the ansatz of Slater determinant of exact Hamiltonian.

By using the Lagrangian multiplier method, we minimize the expectation value of the exact Hamiltonian,

δ δφ^∗_α(x)

"

hΦ| ˆH|Φi −

N

X

i=1

ǫi

Z

dyφi(y)φ^∗_i(y)

#

= 0 , (44)

where δφ^∗_α(x) is single-particle wavefunction. Inserting the expectation value of the exact Hamiltonian in second quantization,

hΦ| ˆH|Φi =

N

X

i=1

h i|ˆt|ji +1 2





N

X

i,j=1

hij|ˆv|iji −

N

X

i,j=1

hij|ˆv|jii



 , (45)

into Eq. (44) results to the Hartree-Fock equation,

 ˆt +

Z

dyρ(y)v(x, y)



φα(x) − Z

dyρ(x, y)v(x, y)φα(y) = ǫαφα(x) , (46) where ρ is density function defined as,

ρ(y) =

N

X

i=1

φ^∗_i(y)φi(y) ρ(x, y) =

N

X

i=1

φ^∗_i(y)φi(x) . (47)

By comparing this equation with Eq. (42), we find the Hartree-Fock poten- tial,

ˆ vHF =

Z

dyρ(y)v(x, y)φα(x) − Z

dyρ(x, y)v(x, y)φα(y) . (48) By solving the eigenvalue problem for the Hartree-Fock potential, we obtain the new set of infinite eigenfunctions and eigenvalues. We choose the ground state orbital and repeat the procedure. By iterating this scheme until the pro- cedure converges, we find the eigenfunctions and eigenvalues of the system.

This description is extracted from [6].

2.2.4 Analytical solution

Considering a 2-electron interacting system, the Hamiltonian contains just one interacting term. This is the simplest interacting system, which can be solved accurately.

Consider a 2-electron Hamiltonian,

H =

2

X

i=1

((Pi+^e_cA(ri))²

2m^∗ +1

2m^∗ω²₀r²_i) + e²

4πǫ|r²− r¹| . (49) One approach to solve the eigenvalue problem for this Hamiltonian is decou- pling of the Hamiltonian in the center of mass coordinate system. We change the variables to the center of mass and the relative distance of two electrons as follows,

R=1

2(r1+ r2) r= r1− r², (50) which leads to decoupling of the Hamiltonian,

H = Hcom+ Hrel, (51)

where Hcomis the Hamiltonian for the center of mass and Hrel is the Hamilto- nian corresponding to the relative motion of two electrons,

Hcom=(P +^2e_cA(R))²

2M^∗ +1

2M^∗ω²₀R², (52) Hrel=(p + _2c^eA(r))²

2µ^∗ +1

2µ^∗ω²₀r²+ e²

4πǫr , (53)

where M^∗= 2m^∗ and µ^∗= ¹₂m^∗.

Roughly speaking, the wavefunction of a 2-electron system is like a Fock- Darwin orbital modified by the Hrel. For performing the exact solution, let the ansatz,

Ψ = ξ(R)ρ(r) , (54)

which results to the eigenfunctions, ξ(R) = e^iMΘ

√2π

UM(R)

R^1/2 M = 0, ±1, ±2, . . . , (55) ρ(r) = e^imθ

√2π um(r)

r^1/2 m = 0, ±1, ±2, . . . , (56) where (R, Θ) and (r, θ) are the polar coordinates of the center of mass and the relative vector coordinates, and the radial functions UM(R) and um(r) can be found in standard textbooks [15].

2.3 On the importance of e-e Coulomb interaction

2.3.1 Wigner Crystal

In 1934, E.P. Wigner reports on the role of e-e interaction for a bulk of electron gas. The essence of his idea was that the energy of interacting system decreases due to the e-e interaction, through the modification of the wavefunction. Defin- ing the dimensionless variable,

rs= r0

a0

, (57)

where r0is the interparticle spacing and

a0= ¯h²

mc² , (58)

is the Bohr radius. Energy per particle in the low-density limit can be obtained as,

E N = e²

2a0[−1.79 rs

+2.66 r^3/2s

+ ...] , (59)

where N is the number of electrons [4]. The first term of Eq. (59) is e-e interaction potential, and the second term is kinetic energy. By increasing rs

(decreasing the electronic density), the interaction term dominates, and the kinetic energy will be negligible. This implies that electrons become localized, and build a crystal, known as Wigner crystal.

In many-electron system with geometric symmetry, the non-interacting term of the Hamiltonian results to the symmetric wavefunctions, or delocalized elec- trons. By considering interacting term, this symmetry will be broken, and the electrons will be localized (Wigner crystal). For 2-d QD with circular symmetry, Wigner crystallization happens in high magnetic field, where the e-e interaction dominates.

2.3.2 Fractional quantum Hall effect

Fractional quantum Hall effect (FQHE) is another interesting effect rising from strong e-e interaction. “It implies, that many electrons, acting in concert, can create new particles having a charge smaller than the charge of any individual electron” states H.L. Str¨omer in his Nobel lecture in 1998. Here we follow his intuitive picture of FQHE.

Consider a many-electron circular QD in the presence of a high homogeneous magnetic field, where the Landau level is occupied by the electrons. According to quantum mechanics, electrons are extended over the QD. Therefore, we have charge distribution over the QD. In this picture, the effect of magnetic field on this charge distribution is to building local whirlpools, called vortices. A vortex contains one quantum unit of magnetic flux,

Φ0= hc/e = 2.07 × 10⁻¹⁵W b , (60) and has no charge inside, since it throws out the charge to the edge of the vortex. Vortices are absence of charge, and like electrons, they are spread out uniformly over the QD. The lowest Landau level has D = Φ/Φ0energy states.

In this level, we have same number of magnetic quanta over QD. If the lowest Landau level is fully filled, then for each electron we have one vortex. This is the condition of i=1 for the integral quantum Hall effect (IQHE), which is the result of Pauli exclusion principle.

If we fill up 1/3 of the lowest Landau level (Landau level filling factor ν = 1/3), then for each electron we have 3 vortices. The only way of reduction of e-e interaction is the uniformly spread of the vortices which implies that every 3 vortices surround one electron. The composition of one electron and its surrounding 3 vortices is called composite particle, CP. This virtual particle can have fractional charge (in this case 1/3), and it can be either fermion or boson, depending on its configuration. Fermions have half-integer spin, and the many-fermion wavefunction is anti-symmetric. That is, if we exchange the state of two fermions, the many fermion wavefunction will change the sign. Bosons have integer spin and the many-boson wavefunction is symmetric.

How can we find out whether a CP is fermion, boson or anyon? If we exchange two CPs, the exchange of the electrons reverse the sign of the wave- function, but the exchange of each attached vortex also changes the sign of the wavefunction (It seems that vortices follow fermion statistics). For example, the CP, that we discussed above with 1/3 charge, is a boson, since the wavefunction changes the sign 4 times, one for electron exchange and 3 for the exchange of the 3 attached vortices. So, we can conclude, that a CP consist of a fermion and odd/even number of vortices is a boson/fermion.

Interesting enough, that the concept of composite particle reduces many- particle system in the presence of a homogeneous magnetic field, to single- particle system in the absence of an external magnetic field. This is due to the fact, that the magnetic field is taken into account by defining the magnetic quanta, and the interparticle interaction is taken into account by displacement of vortices, in a way that minimizes the Coulomb interaction. Even more, by

tuning the magnetic field, we can control the number of vortices, and conse- quently, the type of the system, to obtain many-fermion or many-boson system.

3 Model and Implementation

We consider a system with N electrons trapped in a harmonic confinement in the presence of a homogeneous magnetic field. We have modeled this system as the following Hamiltonian,

H =

N

X

i=1

((Pi+^e_cA(ri))²

2m^∗ +1

2m^∗ω²₀r²_i) +

N

X

i6=j=1

e²

8πǫ|r^j− rⁱ| . (61) For having an idea about the role of magnetic field, we can write the Hamil- tonian in terms of cyclotron frequency,

H =

N

X

i=1

(Pi2

2m^∗ −1

2ωcLi+1 2m^∗(ω_c²

4 + ω²₀)r_i²) +

N

X

i6=j=1

e²

8πǫ|r^j− rⁱ| . (62) To see the effect of e-e interaction in energy spectrum, we need the exact solution of the system. For this purpose, we have considered the exact diago- nalization of the Hamiltonian in second quantization formalism. Hamiltonian in second quantization can be written as,

H =ˆ X

rt

Trtc^†_rct+1 2

X

pqrs

Vpqrsc^†_pc^†_qcscr, (63) where Trt is the matrix elements of single-particle operator, known as Fock- Darwin spectrum. The single-particle states are basically eigenstates of the Fock-Darwin Hamiltonian, which are known to us. By having single-particle eigenstates, matrix elements of the two-particle operator,Vpqrs, can be calculated by Eq. (30).

The next step is to build up Fock space basis. Since the total angular momentum, ˆL =Pne

i=1lˆi, commutes with the Hamiltonian,

[ ˆH, ˆL] = 0 , (64)

they have common eigenstates, and the energy levels can be labeled by quantum number L =Pne

i=1li. Consequently, we can choose basis by fixing the quantum number L. Since we want to diagonalize the Hamiltonian matrix, we need to have a finite matrix. So therefore, we need to cut out the Hilbert space of the single-particle system, and this costs us losing the accuracy of the energy values;

particularly, for large number of particles and for high energy states. However, we can approach exact values by choosing the states that contribute more in energy values. The low lying states in the Fock-Darwin spectrum can be a good candidate.

In our calculations, we have considered 36 low lying Fock-Darwin single- particle states, to construct Fock space basis. Another aspect that we should consider is that the z component of the total spin, Ms, is fixed. This is due to this fact that we do not consider any spin interaction in our model. This leads us to considering Pauli exclusion principle as another guide for constructing the Fock space basis.

To give an example of construction of the finite Fock space basis, we consider 2-electron system with Ms= 0. By considering 36 lowest single-particle states, and choosing total angular momentum L = 0, we can construct Fock space basis consisting of 104 states. Some of the states are shown in following table. The single-particle state is labeled as the quantum numbers (n, l),

(0, 0)(0, 0) (0, 1)(0, −1) (0, 2)(1, −2) (0, −3)(2, 3) (0, 0)(1, 0) (0, 1)(1, −1) (0, 2)(2, −2) (0, 3)(0, −3) (0, 0)(2, 0) (0, 1)(2, −1) (1, 0)(0, 0) (0, 3)(1, −3) (0, 0)(3, 0) (0, 1)(3, −1) (1, 0)(1, 0) (0, 3)(2, −3) (0, −1)(0, 1) (0, −2)(0, 2) (1, 0)(2, 0) (1, −1)(0, 1) (0, −1)(1, 1) (0, −2)(1, 2) (1, 0)(3, 0) (1, −1)(1, 1) (0, −1)(2, 1) (0, −2)(2, 2) (0, −3)(0, 3) (1, −1)(2, 1) (0, −1)(3, 1) (0, 2)(0, −2) (0, −3)(1, 3) (1, −1)(3, 1)

By using this basis, we build up the Hamiltonian with the dimension of 104.

To reach to the final point, finding energy spectrum and energy states, we need to diagonalize the Hamiltonian. This has been done by using subroutines based on Lanczos method.

The exact diagonalization has been performed for a few-electron system up to 4 electrons, and the energy spectrum have been obtained. The results are in atomic units, and the default values for parameters are confinement frequency, ω0= 1 and effective mass of electron, m^∗_e= me= 1. In the case of using other values, it will be mentioned.

4 Numerical results and discussion

4.1 Two-electron system

The spectrum of the non-interacting 2-electron system is shown in Fig. 2. The ground state, which belongs to L=0, reflects the effect of quadratic term of the Hamiltonian. The behavior of the energy lines for the excited states is due to the competition between quadratic and linear term in the Hamiltonian, Eq. (62).

In the high magnetic field, the convergence to the lowest Landau level can be seen, which is due to the negligible confinement potential compared to the high magnetic field.

Now, what happens in the presence of e-e interaction? Our result, for the interacting 2-electron system with the Hamiltonian presented in Eq. (51), is depicted in Fig. 3. This result is in agreement with analytical result done by

E. Taut, Fig. 4. The deformation of the spectra lines (particularly low lying energies), due to the e-e interaction, leads to the crossing of the lines and the new degeneracies. By changing the magnetic field, the transition to the higher orbitals occurs. To explain this transition qualitatively, we focus on the effect of magnetic field on orbitals. By increasing the magnetic field, the orbitals shrink, and the e-e repulsion increases. In a critical point (the crossing or degeneracy point), the ground state favors the transition to the outer orbital labeled by total angular momentum L=-1, where the e-e repulsion is less. To keep increasing the magnetic field leads to the another orbital transition, and so on (Fig. 5).

The e-e interaction as a function of magnetic field is plotted in Fig. 6. Compare this result with the analytical result published by D. Pfannkuche, et al (Fig. 7).

So far we discussed about the orbital part of spin-wave function, but the question is, how magnetic field affects the total spin. The calculation shows a singlet-triplet transition of total spin related to the orbital transition, Fig. 5.

This was expected due to the Pauli exclusion principle. The total wavefunction of many-fermion system must be anti-symmetric and since the even(odd) total angular momentum is symmetric(anti symmetric), its spin function should be antisymmetric(symmetric). This argument can be confirmed by considering another case, when a 2-electron system has total z component spin Ms = 1, Fig. 8. As we see, the ground state belongs only to the odd total angular momenta, which are antisymmetric, and this is due to this fact that the only choice for the symmetric spin function is triplet.

To investigate the role of confinement energy, the e-e interaction for the different values of confinement frequency, ω0, are plotted in Fig. 10-11. The comparison of these two graphs shows the strong response of e e-e interaction to confinement frequency for a low magnetic field. This was expected, because in the absence or low effect of magnetic field, the effective radius of the orbitals depends on the confinement potential. However, by increasing the magnetic field, the orbitals shrink and the effect of confinement weakens. In the high magnetic field, the role of confinement potential vanishes, because orbitals are concentrated in the center and do not feel the edges of the QD.

Another effect of e-e interaction can be seen in the behavior of energy levels belonging to the lowest Landau level. By increasing the magnetic field, energy lines converge and after a critical point they start to split. This splitting in very high magnetic field is shown in Fig. 9, which is just the signature of e-e interaction.

4.2 Few-electron system

Numerical result for 3-4 electrons with different total spin have been obtained.

In 3-electron system, with Ms= 1/2, the transition to higher angular momen- tum occurs, Fig. 13, which leads to the oscillation of the spin state between doublet and quartet, Fig. 14. In 3-electron system, with Ms = 3/2, the only choice of spin state is quartet, so therefore we observe just anti-symmetric or- bitals, Fig. 13. The energy spectrum of non-interacting 3-electron systems are given in Fig. 12 and Fig. 15.

The energy spectrum of spin polarized 4-electron systems with different total spin are plotted in Fig. 17-20.

5 Conclusion

We have considered systems consisting of a few electrons trapped in a 2-d circu- lar quantum dot with a harmonic confinement and in the presence of an external homogeneous magnetic field. The traditional way of solving the Schr¨odinger equation for the systems with more than 2 electrons is impossible due to the interacting term in the Hamiltonian. Since the approximation methods can not reveal the effect of e-e interaction, we have applied the exact diagonalization method.

Exact diagonalization is a method that diagonalizes the second quantization Hamiltonian. In general, the Hamiltonian of an interacting system consists of single-particle and two-particle operators. In second quantization the matrix elements of these operators appear as coefficients of creation and destruction operators. To write the Hamiltonian in second quantization, we need to build Fock basis based on single-particle eigenstates. For this we need to cut the Hilbert space into a finite space, which costs the reduction of precision of this method particularly for big number of particles and high excited levels. In the other hand, the size of the Hamiltonian increases exponentially with the number of particles, which restricts us to systems up to 6-7 particles.

The diagonalization of Hamiltonian can be performed by using subroutines based on Lanczos method. The idea behind this method is that by starting with an arbitrary vector and through a minimization process, we build a complete set of orthonormal vectors. The matrix in these basis will be tridiagonal, which is convenient for diagonalization.

In this project, we have diagonalized the Hamiltonian for 2, 3, and 4-electron systems. The low lying energy levels have been presented, which show the effect of e-e interaction. By increasing magnetic field, the transition to bigger orbitals occurs, which is the consequence of the strong e-e repulsion. This transition, in some systems, results to the change of the total spin due to the antisymmetric property of the total wavefunction. Singlet-triplet oscillation in 2-electron system with Ms = 0 and doublet-quartet transition in 3-electron system with Ms= 1/2 are some examples of this effect.

The e-e interaction gives rise to another interesting effect, namely FQHE.

The results show the splitting of the lowest Landau level in high magnetic field, which indicates the strong e-e interaction in this level. Due to this fact, electrons will be localized and make Wigner crystal. For the partially filled Landau level, this leads to the FQHE.

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List of Figures

1 Single-particle energy levels (Fock-Darwin spectrum) [2]. . . 26 2 The low lying energy levels for non-interacting 2-electron system

with total z component of spin Ms = 0 (one spin-up and one spin-down electron). Total angular momentum L=0 is ground state. . . 26 3 The low lying energy levels for interacting 2-electron system with

total z component of spin Ms = 0 (one spin-up and one spin- down electron). By increasing the magnetic field, the transition to higher angular momentum occurs. . . 27 4 The low lying energy levels for interacting 2-electron system with

Ms= 0, obtained by analytical calculation [15]. . . 27 5 The transition to higher angular momentum and singlet-triplet

oscillation by changing of the magnetic field for 2-electron system with Ms= 0. . . 28 6 e-e interaction of the ground state of 2-electron system with Ms=

0. Increasing of the e-e interaction leads to the singlet-triplet oscillation which decreases the total energy of the system. . . 28 7 Difference between the interacting and non-interacting ground

state energy of quantum dot helium [10]. . . 29 8 The low lying energy levels for interacting 2-electron system with

total z component of spin Ms = 1 or Ms = −1. Since the spin state is triplet, we observe only the transition between odd (an- tisymmetric) angular momenta. . . 30 9 The splitting of the lowest Landau level for interacting 2-electron

system with Ms= 0 is compared with the non-interacting system. 30 10 e-e interaction of the low lying energy levels (L=-5,. . . ,5) for 2-

electron system with Ms= 0 and ω0= 1. . . 31 11 e-e interaction of the low lying energy levels (L=-5,. . . ,5) for 2-

electron system with Ms= 0 and ω0= 2. . . 31 12 The low lying energy levels for non-interacting 3-electron system

with Ms= 1/2 (or Ms= −1/2). . . 32 13 The low lying energy levels for interacting 3-electron system with

Ms= 1/2 (or Ms= −1/2). . . 32 14 The transition to higher angular momentum and doublet-quartet

transition by changing of the magnetic field for 3-electron system with Ms= 1/2. . . 33 15 The low lying energy levels for non-interacting 3-electron system

with Ms= 3/2 (or Ms= −3/2). . . 33 16 The low lying energy levels for interacting 3-electron system with

Ms= 3/2 (or Ms= −3/2). . . 34 17 The low lying energy levels for non-interacting 4-electron system

with Ms= 0. . . 34 18 The low lying energy levels for interacting 4-electron system with

Ms= 0. . . 35

19 The low lying energy levels for non-interacting 4-electron system with Ms= 1 (or Ms= −1). . . 35 20 The low lying energy levels for interacting 4-electron system with

Ms= 1 (or Ms= −1). . . 36

Figure 1: Single-particle energy levels (Fock-Darwin spectrum) [2].

2 4 6 8 10 12

0 2 4 6 8 10

Energy(hartree)

ω_c

Figure 2: The low lying energy levels for non-interacting 2-electron system with total z component of spin Ms = 0 (one spin-up and one spin-down electron).

Total angular momentum L=0 is ground state.

2 4 6 8 10 12

0 2 4 6 8 10

Energy [hartree]

ω_c

L=5 L=4 L=3 L=2 L=1 L=0 L=-1 L=-2 L=-3 L=-4 L=-5

Figure 3: The low lying energy levels for interacting 2-electron system with total z component of spin Ms = 0 (one spin-up and one spin-down electron).

By increasing the magnetic field, the transition to higher angular momentum occurs.

Figure 4: The low lying energy levels for interacting 2-electron system with Ms= 0, obtained by analytical calculation [15].

0 2 4 6 8 10

0 2 4 6 8 10

Total angular momentum / Spin

ω_c Angular momentum

spin

Figure 5: The transition to higher angular momentum and singlet-triplet oscil- lation by changing of the magnetic field for 2-electron system with Ms= 0.

1 1.05 1.1 1.15 1.2

0 2 4 6 8 10

Energy(hartree)

ω_c

Singlet Triplet Singlet Triplet

Figure 6: e-e interaction of the ground state of 2-electron system with Ms= 0.

Increasing of the e-e interaction leads to the singlet-triplet oscillation which decreases the total energy of the system.

Figure 7: Difference between the interacting and non-interacting ground state energy of quantum dot helium [10].

3 4 5 6 7 8 9 10 11 12

0 2 4 6 8 10

Energy(hartree)

ω_c

L=5 L=4 L=3 L=2 L=1 L=0 L=-1 L=-2 L=-3 L=-4 L=-5

Figure 8: The low lying energy levels for interacting 2-electron system with total z component of spin Ms = 1 or Ms = −1. Since the spin state is triplet, we observe only the transition between odd (antisymmetric) angular momenta.

30 32 34 36 38 40 42 44

30 32 34 36 38 40

Energy(hartree)

ω_c

non-interacting interacting

Figure 9: The splitting of the lowest Landau level for interacting 2-electron system with Ms= 0 is compared with the non-interacting system.

0 0.5 1 1.5 2 2.5

0 2 4 6 8 10

e-e interaction energy [hartree]

ω_c

Figure 10: e-e interaction of the low lying energy levels (L=-5,. . . ,5) for 2- electron system with Ms= 0 and ω0= 1.

0 0.5 1 1.5 2 2.5

0 2 4 6 8 10

e-e interaction energy [hartree]

ωc

Figure 11: e-e interaction of the low lying energy levels (L=-5,. . . ,5) for 2- electron system with Ms= 0 and ω0= 2.

4 6 8 10 12 14 16

0 2 4 6 8 10

Energy(hartree)

ω_c

Figure 12: The low lying energy levels for non-interacting 3-electron system with Ms= 1/2 (or Ms= −1/2).

6 8 10 12 14 16 18 20

0 2 4 6 8 10

Energy(hartree)

ωc

Figure 13: The low lying energy levels for interacting 3-electron system with Ms= 1/2 (or Ms= −1/2).

0 2 4 6 8 10

0 2 4 6 8 10

Total angular momentum / Spin

ω_c Angular momentum

spin

Figure 14: The transition to higher angular momentum and doublet-quartet transition by changing of the magnetic field for 3-electron system with Ms= 1/2.

6 8 10 12 14 16 18 20

0 2 4 6 8 10

Energy(hartree)

ωc

Figure 15: The low lying energy levels for non-interacting 3-electron system with Ms= 3/2 (or Ms= −3/2).

8 10 12 14 16 18 20

0 2 4 6 8 10

Energy(hartree)

ω_c

Figure 16: The low lying energy levels for interacting 3-electron system with Ms= 3/2 (or Ms= −3/2).

6 8 10 12 14 16 18 20

0 2 4 6 8 10

Energy(hartree)

ωc

Figure 17: The low lying energy levels for non-interacting 4-electron system with Ms= 0.

10 11 12 13 14 15 16

0 1 2 3 4 5

Energy(hartree)

ω_c

Figure 18: The low lying energy levels for interacting 4-electron system with Ms= 0.

6 8 10 12 14 16 18 20

0 2 4 6 8 10

Energy(hartree)

ωc

Figure 19: The low lying energy levels for non-interacting 4-electron system with Ms= 1 (or Ms= −1).

10 10.5 11 11.5 12 12.5 13 13.5 14

0 0.5 1 1.5 2 2.5 3

Energy(hartree)

ω_c

Figure 20: The low lying energy levels for interacting 4-electron system with Ms= 1 (or Ms= −1).